Let Me Float in Fluids of Your **** and Be Born Again as Us
Is there a libation object in the sky than Saturn? Maybe Jupiter, but I similar Saturn a little better. If y'all know what y'all are looking for, yous tin can run into the rings of Saturn fifty-fifty with a pair of binoculars. Personally, I love getting out the telescope and showing Saturn to people. Their expressions when they run into information technology testify their amazement. Most people don't realize that you tin can ACTUALLY run across the rings.
Plenty about viewing Saturn. Hither is the one affair that bothers me. You will ofttimes come across in textbooks and other media that Saturn has a low density and would really float in h2o. No. This is wrong. Well, information technology's kind of wrong.
The Density of Saturn
Let'south assume that Saturn is a sphere. Nosotros can easily calculate the density now. Well, easily assuming that we look up values for the radius and the mass. Co-ordinate to Wikipedia, Saturn has a mass of 5.68 x 1026 kg and a radius of about five.6 x ten7 meters. Knowing the volume of a sphere, we go the following adding for the density.
The density of water is 1000 kg/thouthree. What does this mean? Well, if I accept a block of some material underwater then I tin can draw the following two forces on it:
On the surface of the Earth, the magnitude of the gravitational forcefulness can be written as:
Here I just wrote the mass of the object as the product of the density of the object (ρo) and the volume of the object (Five o). For the buoyancy strength, I tin calculate this as the weight of the h2o displaced. This would be written as:
Both the weight and the buoyancy force have the same V o grand term. The only thing that is unlike is the density. So, if the density of water is greater than the density of the object, the buoyancy force when the object is fully submerged will be greater than the weight. In order to be at equilibrium, the object would be just partially submerged. Nosotros commonly call this "floating". And here you see that if the density of an object is less than the density of water, that object will float.
If you want a more detailed derivation of the buoyancy forcefulness - cheque out this mail nigh the Magdeburg Water Bridge.
Would Saturn Float?
Saturn'south density is less than water. Things with a density less than h2o bladder - things like ducks, tiny rocks and gravy. So information technology seems logical that Saturn would also float. Right? Incorrect.
How much water would you demand for Saturn to bladder? Let'southward assume for at present that this is some ginormous planet with as much water equally nosotros need. Too, I will presume that in this region of water, the gravitational field is constant and pointing direct down since the planet is then large.
If the planet could float (see below), how deep would the water need to exist? For a floating object, the buoyancy force is equal to the gravitational forcefulness. This means that but part of the planet would be underwater. Only how much? If I telephone call the volume of the planet underwater 5 d (d is for displacement), so I can write:
This means that the volume of water displaced will be the book of Saturn multiplied by the ratio of the densities. Using my density of Saturn, 77.2% of information technology would exist underwater. How deep would this be? Hither is a film.
You lot tin see I need to find the value for h which is the depth the planet would become underwater. It's clear it will be larger than the radius of the planet, but by how much? Instead of deriving the formula for the volume of a fractional sphere - I will use this Wikipedia page for a spherical cap. This says that the volume of the cap (the top part) would be:
If I ready this cap volume to 0.228 the volume of the full sphere, then I tin can solve for a. I will skip the details - you can do this for a homework trouble if you like. It'southward non besides hard to solve, but I become a value for a of 0.6189*R. This means that h would be 1.38*R. With the radius of Saturn, y'all would need water that is 7.7 x 107 meters deep. Maybe you lot would like this depth in unlike units. How about a water depth of six Earth diameters?
Permit me make a sketch of this. I am just going to draw a water planet that looks big enough to be more often than not "flat" around our floating Saturn.
I left the within of the planet empty - I don't know why. Yet, based on this sketch the h2o surface planet would take a radius 8 times larger than Saturn's radius. This makes the water planet on the same order as the size of the Sun - except h2o. Water is hydrogen and oxygen. You know what else has a lot of hydrogen? Yes, the Sun. I oasis't done the calculations, but it seems like a planet the size of our water planet would have plenty pressure level in the cadre to start nuclear fusion.
Oh, so that'southward why I made it hollow. Still, the pressure level at the bottom of this ocean would be way too high for the stuff at the lesser to nonetheless be liquid water. Really, I don't know what would happen to it. I merely don't think y'all could make whatsoever body of water this deep no matter what you try.
Saturn Still Wouldn't Float
Ok, perchance you found some awesome way to make water actually really deep but nevertheless water. Mayhap you have devoted the resources of the unabridged solar arrangement just to brand a giant ocean of water. Ok, I become information technology. Still Saturn wouldn't bladder.
If you take a ping pong ball and toss it in your tub, it will float. A ping pong ball is a rigid object. Saturn is not rigid. The large bulk of Saturn'due south outer book is filled with molecular hydrogen. The interior is something much denser - maybe metallic hydrogen and/or a rocky core. The denser materials are in the center considering a gravitational interaction. If you lot like, you could think of the collective gravitational force of all the bits of Saturn pulling such that the denser stuff is in the middle supporting the lower density materials.
But what would happen if y'all put this non-rigid object on the giant water planet? If the planet is very very large in mass, the net gravitational field will be towards the center of the h2o planet and not towards the centre of Saturn. This means that all of that material - especially the rocky cadre volition also exist pulled to the center of planet water. Allow me change my floating Saturn diagram to bear witness the core.
What is going forces are going to be acting on the core? Well, at that place the gravitational force of planet water pulling on it. Simply what pushes upward on information technology? The hydrogen in the atmosphere of Saturn pushes up - but not very much, it's just non dense enough. That means this cadre will "fall" towards the surface of the water planet. The hydrogen temper will and then motility up and probably become part of the water planet's temper. This would kind of be like trying to hold a raw egg without the shell. It but doesn't stay together.
In the stop, you lot would have a giant rocky core at the lesser of the water planet's sea. If y'all want to call the destruction of a planet "floating", well I guess that's ok. Or maybe we could keep the one-time definition of floating and leave Saturn where information technology is.
Then, what should you say about Saturn'due south density? How about something like this:
Oh, I think I should talk most how humans find the mass and volume of Saturn. That volition exist another post though.
Don't forget that today (July nineteen, 2013) is Smiling and Wave at Saturn 24-hour interval. At around 21:30 UTC, the Cassini spacecraft will accept a movie of Earth and Saturn at the aforementioned fourth dimension. So, wave and rummage your hair.
Source: https://www.wired.com/2013/07/no-saturn-wouldnt-float-in-water/
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